Older type Fenders as well as new RI's and the 60's Classic (which I have) all tend to be a bit noisy, nature of the beast! But if you are playing on a stage where the equipment (sound and lighting) are not properly grounded, you will have problems.

When I was on the road, my band had two electrical boxes that we would tap into the clubs main (when we could). This would isolate the power used for the lights from the power for all the sound equipment. Very rarely did we have buzzing problems.

Now I had the same problem as you about a week ago. Loud buzz coming from the amp when I was not touching the bass.

Solution: Shut off the neon Bud sign behind the stage. No more problem!

If you have this problem even at home (or in other places), you might have a loose ground wire. There should be a ground wire attach to underside of the bridge. If this comes loose, you Will have a buzzing problem. If that is the case, just remove the bridge and re-attach the ground wire. (Also you could check in the control cavity to see if anything is loose there)

ALSO, it could be a grounding problem with your amp, try some other basses and see if it does the same thing.....